nLab differential algebroid

Contents

Context

Enriched category theory

Algebra

Categorification

Contents

Idea

A differential algebroid is an oidification of the concept of differential algebra.

Definitions

Let KK be a commutative ring and CC be a KK-linear category. Then CC is a KK-differential algebroid if for every hom-KK-module Mor(a,b)Mor(a,b) there is a KK-linear morphism d Mor(a,b):Mor(a,b)Mor(a,b)d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b) such that for all objects a,b,cOb(C)a,b,c\in Ob(C), morphisms f:Mor(a,b)f:Mor(a,b) and g:Mor(b,c)g:Mor(b,c), and KK-linear morphisms d Mor(a,b):Mor(a,b)Mor(a,b)d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b), d Mor(b,c):Mor(b,c)Mor(b,c)d_{Mor(b,c)}:Mor(b,c) \to Mor(b,c), and d Mor(a,c):Mor(a,c)Mor(a,c)d_{Mor(a,c)}:Mor(a,c) \to Mor(a,c), a generalised Leibniz rule is satisfied:

d Mor(a,c)(gf)=d Mor(b,c)(g)f+gd Mor(a,b)(f). d_{Mor(a,c)}(g \circ f) = d_{Mor(b,c)}(g) \circ f + g \circ d_{Mor(a,b)}(f) \,.

If all three objects are the same, this reduces down to the Leibniz rule for a derivation.

Examples

algebraic structureoidification
truth valuepreorder
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
anti-involutive monoiddagger category
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
cancellative monoidcancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
monoidal poset2-poset
strict monoidal groupoid?strict (2,1)-category
strict 2-groupstrict 2-groupoid
strict monoidal categorystrict 2-category
monoidal groupoid(2,1)-category
2-group2-groupoid/bigroupoid
monoidal category2-category/bicategory

Last revised on May 23, 2021 at 17:10:36. See the history of this page for a list of all contributions to it.